A surveillance camera rotates through an angle of 117^circ during a pan. What is the measure of this angle in...

1. TRANSLATE the problem information

• Given information:
  • Angle measure: \(117\) degrees
  • Need to convert to radians
  • Express answer in terms of \(\pi\) as a reduced fraction

• This tells us we need to use the degree-to-radian conversion

2. TRANSLATE the conversion setup

• Use the fundamental relationship: \(180° = \pi\) radians
• Set up the conversion: angle in radians = \(\text{degrees} \times (\pi/180°)\)
• Substitute our value: \(117° \times (\pi/180°) = (117\pi/180)\)

3. SIMPLIFY the fraction to reduced form

• Find the GCD of 117 and 180:
  • \(117 = 9 \times 13\)
  • \(180 = 9 \times 20\)
  • \(\mathrm{GCD}(117, 180) = 9\)

• Divide both numerator and denominator by 9:
  • \(117 \div 9 = 13\)
  • \(180 \div 9 = 20\)

• Final simplified form: \(13\pi/20\)

Answer: \(13\pi/20\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may set up the conversion incorrectly, using \((180/\pi)\) instead of \((\pi/180)\) as the conversion factor.

This leads to calculating \(117 \times (180/\pi) = 21060/\pi\), which doesn't match the required format and produces a decimal approximation rather than a fraction in terms of \(\pi\).

This causes confusion since the answer doesn't match any reasonable expectation and leads to guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \((117\pi/180)\) but fail to reduce the fraction properly.

They might not find the correct GCD, perhaps identifying it as 3 instead of 9, leading to an answer like \(39\pi/60\), or they might make arithmetic errors during the division process.

This may lead them to select an incorrect answer choice or submit an unreduced fraction.

The Bottom Line:

This problem requires precise execution of both the conversion setup and fraction reduction. Students must remember the correct conversion factor and systematically find the GCD to achieve the required reduced form.